A braid on $n$ strands is a collection of $n$ curves in Euclidean space $E={ℝ}^{3}$ starting at the $n$ points $\left(j,0,1\right)$ and ending at the $n$ points $\left(j,0,0\right)$ for $j\in \left\{1,\mathrm{...},n\right\}$, each meeting the planes $z=c$, with $c\in \left[0,1\right]$, in a unique point.

A physical braid

In particular, the curves, referred to as strands, are disjoint and descend monotonically.

Just like knots and links, braids are considered equal when they are isotopic. At each instance of a smooth deformation realizing the isotopy, the braid should remain a braid; in particular, the end points of the braid remain fixed.

The diagram of a braid on 4 strands

Just like knots and links, we will depict braids by means of diagrams; in this case we take the projections to be orthogonal onto a vertical plane behind the braid. Crossings are treated as for links, with overpasses and underpasses. Because of our restrictions on braids, no Reidemeister I move will be realizable. Reidemeister II and III moves are possible. For instance, in the above diagram, a sequence of two Reidemeister II moves shows that the corresponding braid is (isotopic to) the braid pictured below.

The elementary braid ${s}_{1}$ on $4$ strands

The strands in the braid are numbered 1,...,$n$ from left to right, so that the leftmost strand is always strand 1 (no matter what position it started in).

The elementary braid ${s}_{i}$ is the braid interchanging the two strands numbered $i$ and $i+1$, where the former underpasses the latter and all other strands go straight from top to bottom.

The inverse of the elementary braid ${s}_{i}$ is the braid, denoted ${\left({s}_{i}\right)}^{-1}$, that interchanges the two strands numbered $i$ and $i+1$, where the former overpasses the latter and all other strands go straight from top to bottom. We will soon see why it makes sense to speak of an inverse here.

Two braids, say $b$ and $c$, on $n$ strands can be composed into a single braid, denoted $b·c$, by placing $c$ under $b$, thereby identifying the top ends of $c$ with the bottom ends of $b$, then viewing these identified points as ordinary points on the resulting longer strands, and scaling so that the top ends of $b$ stay in place and the bottom ends of $c$ are placed at $\left(0,j,0\right)$.

This composition turns the set of all braids on $n$ strands into a group, denoted ${B}_{n}$. For, the composition is clearly associative, the braid all of whose strands go straight down is the identity element with respect to the composition, and the inverse of a braid is obtained by reflecting its diagram in the horizontal line at the bottom of the diagram (and moving the diagram by a translation so as to put the end points back to their standard positions). The latter follows from applications of Reidemeister II moves to find an isotopy from the product of the braid and its inverse to the trivial braid.

The product in ${B}_{4}$ of the braids ${s}_{3}$ and ${s}_{2}$

The group inverse of the elementary braid ${s}_{i}$ now coincides with ${\left({s}_{i}\right)}^{-1}$, as defined in the previous section; this explains the terminology chosen there.

The inverse of the braid ${s}_{3}$

Every braid is (isotopic to) the product of elementary braids. The following result by Emil Artin even characterizes when two diagrams represent the same braid.

Proof

It is easy to deduce that the relations should hold; see the two pictures below.

The braid relation ${s}_{2}·{s}_{1}·{s}_{2}={s}_{1}·{s}_{2}·{s}_{1}$

The braid relation ${s}_{1}·{s}_{3}={s}_{3}·{s}_{1}$

It is harder to see that these relations give a presentation, that is, that every relation is a consequence of these. This was first proved by Emil Artin.

QED

As a consequence of the homogeneity of the defining relations for ${B}_{n}$, we can define a homomorphism of groups $h$ : ${B}_{n}\to \mathbb{Z}$ by decreeing $h\left({s}_{j}\right)=1$. This implies that we can talk about the height of a braid . (Its absolute value is a lower bound on the number of crossings needed to draw it.) The writhe of the closure of a braid coincides with the height of the braid.

A theorem by Alexander shows that every oriented link is equivalent to one obtained by identifying starting and ending points above each other in a suitable braid.

Let $b$ be a braid. The closure of $b$ is the oriented link obtained by connecting each endpoint $\left(j,0,0\right)$ with its vertical counterpart $\left(j,0,1\right)$ by means of an unknotted straight strand that does not cross any of the other strands. The result is a link. Orienting all strands of $b$ downward, we find an oriented link.

The closure of the braid ${s}_{2}·{s}_{1}·{s}_{2}$

The closure of the identity element of ${B}_{n}$ is the disentangled union of $n$ unknots. This shows that it is important to specify the number of strands of the braid.

The closure of a braid is a uniquely determined oriented link. Different braids may represent the same oriented link.

Let $b$ be a braid. We distinguish the following two kinds of Markov move.

The reverses of these moves are also counted as Markov moves.

Alexander realized that every oriented knot can be written as the closure of a braid. We represent an algorithm that, when given an oriented link diagram, changes the diagram without changing the corresponding oriented link, so that it becomes the closure of a braid. This algorithm is due to Vogel.

The major steps in the algorithm are

HOMEWORK CHALLENGE 2 (3 pts): Show termination of this algorithm.

Here is an example.

Vogel's algorithm applied to a link

The resulting braid in the example is ${\left({s}_{1}\right)}^{-1}·{\left({s}_{2}\right)}^{-1}·{\left({s}_{3}\right)}^{-1}·{\left({s}_{2}\right)}^{-2}·{s}_{1}·{\left({s}_{2}\right)}^{-1}·{s}_{3}·{\left({s}_{2}\right)}^{-1}$.

The Vogel algorithm, live in GAP:

Another example of the application of Vogel moves is shown in the sequence of pictures below:

The implementation of this algorithm by Dan Roozemond is also available online . Seifert surfaces can be visualized with Seifertview , by Jack van Wijk . The above braid is represented in Seifertview by ABCBBaBcB.

The Burau representation of the braid group ${B}_{n}$ on $n$ strands is determined by the following images of its generators. The $i$-th generator ${s}_{i}$ of ${B}_{n}$ is mapped to the square matrix of size $n$ of the following form: all of its rows and columns are as in the identity matrix except for the $i$-th and the ($i+1$)-st; the square submatrix of size 2 with entries in $\left\{i,i+1\right\}$ is

The representation ${r}_{n}$ is called the Burau representation of ${B}_{n}$. The restriction of ${r}_{n}$ to $W$ is called the reduced Burau representation of ${B}_{n}$, and denoted ${r\text{'}}_{n}$.

As a consequence, the map sending a braid to the characteristic polynomial of its image under ${r}_{n}$, or any of its coefficients, is invariant under conjugacy, and hence under one of the two Markov moves. By suitably scaling the traces for varying $n$ we can accommodate for the other Markov move and obtain link invariants.

We apply this principle to the characteristic polynomial of the representation on $W$. Note that $W$ is a free module over $\mathbb{Z}\left[t,{t}^{-1}\right]$ with basis the column vectors ${\left(0,...,0,t,-1,0,...,0\right)}^{T}$.

Let $L$ be a link. The Alexander polynomial of a braid $b$ on $n$ strands is the Laurent polynomial

where ${r\text{'}}_{n}$ is the reduced Burau representation of ${B}_{n}$.

Conway observed that the Alexander polynomial can be computed by means of a simple recursion. Consider the initial rule

where ${L}_{+},{L}_{-},{L}_{0}$ are related as follows to an oriented link diagram $L$ in which $c$, say, is a crossing. We let ${L}_{0}$ be $L$ with $c$ replaced by two non-crossing strands. If the sign of $c$ is positive, then we take $L={L}_{+}$ and we let ${L}_{-}$ be $L$ with $c$ replaced by a negative crossing. Similarly, if the sign of $c$ is negative.

Three variations at a crossing of an oriented link $L$

Note that, using the recursive rule appropriately, one can get down from any link to a link with no crossings, but possibly consisting of more than one component. Some additional applications of the rule (in the other direction, creating new crossings) may be required to finish the process.

Call the polynomial obtained from the above recursion the Conway polynomial of the oriented link $L$.

Example 2. 

The picture below shows the stages of a computation of the Conway polynomial of the trefoil.

The stages of a Conway polynomial computation